whenever \(W:L^\infty \rightarrow L^\infty \) , \(W^*:L^1\rightarrow L^1\) . Yajima obtains similar results on \(W^{k,p}\)  assuming more regularity on \(V\) 
(the amount of regularity depends on \(k\) ).
In view of our discussions of the role of zero energy resonances for local decay, it follows
that Yajima's result [1]} can only hold under the assumption that zero energy is neither a resonance nor an eigenvalue.
In three dimensions [1]} requires \(|V(x)|\lesssim \langle x\rangle ^{-\sigma }\)  with \(\sigma >5\)  and therefore improves on [3]}.
